EP2228716B1 - Fehlerbeständige Berechnungen auf elliptischen Kurven - Google Patents

Fehlerbeständige Berechnungen auf elliptischen Kurven Download PDF

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EP2228716B1
EP2228716B1 EP10155001A EP10155001A EP2228716B1 EP 2228716 B1 EP2228716 B1 EP 2228716B1 EP 10155001 A EP10155001 A EP 10155001A EP 10155001 A EP10155001 A EP 10155001A EP 2228716 B1 EP2228716 B1 EP 2228716B1
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elliptic curve
result
integer
fault
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EP2228716A1 (de
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Marc Joye
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Thomson Licensing SAS
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
    • G06F7/60—Methods or arrangements for performing computations using a digital non-denominational number representation, i.e. number representation without radix; Computing devices using combinations of denominational and non-denominational quantity representations, e.g. using difunction pulse trains, STEELE computers, phase computers
    • G06F7/72—Methods or arrangements for performing computations using a digital non-denominational number representation, i.e. number representation without radix; Computing devices using combinations of denominational and non-denominational quantity representations, e.g. using difunction pulse trains, STEELE computers, phase computers using residue arithmetic
    • G06F7/724—Finite field arithmetic
    • G06F7/725—Finite field arithmetic over elliptic curves
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2207/00—Indexing scheme relating to methods or arrangements for processing data by operating upon the order or content of the data handled
    • G06F2207/72—Indexing scheme relating to groups G06F7/72 - G06F7/729
    • G06F2207/7219—Countermeasures against side channel or fault attacks
    • G06F2207/7271—Fault verification, e.g. comparing two values which should be the same, unless a computational fault occurred

Definitions

  • a fault attack introduces an error during cryptographic calculations with the intent to obtain one or more bits of a cryptographic secret, such as a private decryption key.
  • Practical ways to mount fault attacks are surveyed in " The Sorcerer's Apprentice Guide to Fault Attacks” by Hagai Bar-El, Hamid Choukri, David Naccache, Michael Tunstall, and Claire Whelan, Proceedings of the IEEE, 94(2):370-382, 2006 (Earlier version in Proc. of FDTC 2004 ) and in " A Survey On Fault Attacks” by Christophe Giraud and Hugues Thiebeauld, in J.-J. Quisquater, P. Paradinas, Y. Deswarte, and A.A. El Kalam, editors, Smart Card Research and Advanced Applications VI (CARDIS 2004), pages 159-176, Kluwer, 2004 .
  • Adi Shamir provided an elegant countermeasure against fault attacks in " How to Check Modular Exponentiation", presented at the rump session of EUROCRYPT97, Konstanz, Germany, May 13, 1997 . The countermeasure is:
  • r is a 64-bit integer.
  • the correctness of Shamir's method is an application of the Chinese remainder theorem (CRT).
  • CRT Chinese remainder theorem
  • y ' ⁇ y (mod N ) and y ' ⁇ z (mod r ) In the presence of faults, the probability that y ' ⁇ z (mod r ) is about 1/ r .
  • r is a 64-bit value, this means that a fault is undetected with probability of roughly 2 -64 . Larger values for r imply a higher detection probability at the expense of more demanding computations.
  • step 1 CRT denotes an application of the Chinese remainder theorem; namely the so-obtained X satisfies X ⁇ x (mod N ) and X ⁇ 1+ r (mod r 2 ).
  • y ' ⁇ x d (mod N ) and y ' ⁇ (1 +r) d (mod r 2 ) when the computations are not faulty.
  • the correctness of step 3 follows from the binomial theorem.
  • We have 1 + r d ⁇ 0 ⁇ k ⁇ d d k 1 d - k ⁇ r k , where d k denotes the binomial coefficient.
  • the basic operation consists in computing the scalar multiplication kP, that is, P + P + ... + P (k times) where + denotes the group operation on E.
  • a goal of an attacker is to recover the value of k (or a part thereof) by inducing faults.
  • the invention is directed to a method for checking the correctness of a cryptographic operation on a first elliptic curve E(Z / pZ).
  • a processor obtains a third elliptic curve E ⁇ (Z / pr 2 Z) ⁇ E(ZlpZ) x E ( Z / r 2 Z ) given by Chinese remaindering from the first elliptic curve E(Z / pZ) and a second elliptic curve E(Zlr 2 Z), where r is an integer; performs the operation on E ⁇ ( Z/ pr 2 Z) to obtain a first result; performs the operation on E 1 (Zlr 2 Z), where E 1 (Z/r 2 Z) denotes the subset of points in E (Z/ r 2 Z) that reduce modulo r to the identity element on E (Z/ r Z), to obtain a second result; verifies that the first result and the second result are equal in E 1 (Z/ r 2 Z); and if this is the case, outputs
  • a point P ⁇ CRT(P (mod p ), R (mod r 2 )) is formed in E ⁇ (Z/ pr 2 Z) such that P ⁇ reduces to P in E (Z/ p Z), and to R in E 1 (Z/ r 2 Z), where CRT denotes the Chinese remaindering method.
  • the integer r is chosen randomly.
  • the integer r has a predetermined value.
  • integer r is a prime.
  • the elliptic curve is represented as an Edwards curve or as a Jacobi curve.
  • the invention is directed to an apparatus for checking the correctness of a cryptographic operation on a first elliptic curve E (Z/ p Z).
  • the apparatus comprises means for obtaining a third elliptic curve E ⁇ (Z/ pr 2 Z) ⁇ E (Z/ p Z) ⁇ E (Z/ r 2 Z) given by Chinese remaindering from the first elliptic curve E (Z/ p Z) and a second elliptic curve E (Z/ r 2 Z), where r is an integer; performing the operation on Ei (Z/ pr 2 Z) to obtain a first result; performing the operation on E 1 (Z/ r 2 Z), where E 1 (Z/ r 2 Z) denotes the subset of points in E (Z/ r 2 Z) that reduce modulo r to the identity element on E (Z/ r Z), to obtain a second result; verifying that the first result and the second result are equal in E 1 (Z/ r 2 Z); and if this is the case; outputting the first
  • the invention is directed to a computer program product having stored thereon instructions that, when executed by a processor, performs the method according to any of the embodiments of the first aspect.
  • E ( ) denote the set of rational points on elliptic curve E defined over .
  • E 1 (Z/ r 2 Z) ⁇ P in E (Z/ r 2 Z)
  • P modulo r reduces to ⁇ where denotes the identity element on E (Z/ r Z).
  • FIG. 1 is a flow chart illustrating the fault-resistant method according to a preferred embodiment of the invention.
  • the integer r can either be chosen randomly or be fixed to a predetermined value. This integer may also be chosen as a prime. The same holds for point R in E 1 (Z/ r 2 Z) , i.e. it can be chosen randomly or be fixed to a predetermined value.
  • checking step 4 may be performed in several ways; in particular, it can be implemented using so-called infective computation so as to avoid explicit branching instructions.
  • Infective computation is described in " RSA Speedup With Chinese Remainder Theorem Immune against Hardware Fault Cryptanalysis” by Sung-Ming Yen, Seungjoo Kim, Seongan Lim, and Sang-Jae Moon; IEEE Transactions on Computers, 52(4):461-472, 2003 ; (earlier version in Proc. of ICISC 2001 ) and in " Sign Change Fault Attacks On Elliptic Curve Cryptosystems” by Johannes Blömer, Martin Otto, and Jean-Pierre Seifert; in L. Breveglieri, I. Koren, D. Naccache, and J.-P. Seifert, editors, Fault Diagnosis and Tolerance in Cryptography -- FDTC 2006, volume 4236 of Lecture Notes in Computer Science, pages 36-52; Springer-Verlag, 2006 .
  • the elliptic curve may be represented in different ways. Of particular interest are the so-called complete models because the identity element then does not need a special treatment. Examples of such curves are Edwards curves (described by Daniel J. Bernstein and Tanja Lange in “Faster Addition and Doubling on Elliptic Curves", in K. Kurosawa, editor, Advances in Cryptology-ASIACRYPT 2008, volume 4833 of Lecture Notes in Computer Science, pages 29-50; Springer, 2007 ) and Jacobi curves (described by Olivier Billet and Marc Joye in "The Jacobi Model of An Elliptic Curve and Side-Channel Analysis", in M. Fossorier, T. Hoholdt, and A. Poli, editors, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC-15), volume 2643 of Lecture Notes in Computer Science, pages 34-42; Springer, 2003 ).
  • Edwards curves described by Daniel J. Bernstein and Tanja Lange in
  • FIG. 2 illustrates a device according to a preferred embodiment of the present invention.
  • the device 200 comprises at least one interface unit 210 adapted for communication with other devices (not shown), at least one processor 220 and at least one memory 230 adapted for storing data, such as accumulators and intermediary calculation results.
  • the processor 220 is adapted to calculate an exponentiation according to any of the embodiments of the inventive methods, as previously described herein.
  • a computer program product 240 such as a CD-ROM or a DVD comprises stored instructions that, when executed by the processor 220, performs the method according to any of the embodiments of the invention.

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  • General Physics & Mathematics (AREA)
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  • Mathematical Optimization (AREA)
  • Pure & Applied Mathematics (AREA)
  • Computational Mathematics (AREA)
  • Theoretical Computer Science (AREA)
  • Computing Systems (AREA)
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Claims (10)

  1. Verfahren zum Prüfen der Richtigkeit einer kryptographischen Operation auf einer ersten elliptischen Kurve E(Z/pZ), wobei das Verfahren in einem Prozessor (220) die folgenden Schritte umfasst:
    - Erhalten einer dritten elliptischen Kurve E^(Z/pr 2Z) =
    Figure imgb0008
     die durch eine chinesische Restbildung von der ersten elliptischen Kurve E(Z/pZ) und von einer zweiten elliptischen Kurve
    Figure imgb0009
     gegeben ist, wobei r eine ganze Zahl ist;
    - Ausführen der Operation an E^(Z/pr 2Z), um ein erstes Ergebnis zu erhalten;
    - Ausführen der Operation an
    Figure imgb0010
     wobei
    Figure imgb0011
     die Teilmenge der Punkte in
    Figure imgb0012
     bezeichnet, die Modulo r auf das Identitätselement auf
    Figure imgb0013
     reduzieren, um ein zweites Ergebnis zu erhalten;
    - Überprüfen, dass das erste Ergebnis und das zweite Ergebnis in
    Figure imgb0014
     gleich sind; und wenn das der Fall ist;
    - Ausgeben des ersten Ergebnisses der Operation in E^(Z/pr 2Z), Modulo p reduziert.
  2. Verfahren nach Anspruch 1, bei dem die kryptographische Operation eine Skalarmultiplikation in E(Z/pZ) ist, wobei das Verfahren für die fehlerresistente Berechnung von Q = kP auf einer elliptischen Kurve E(Z/pZ) die folgenden Schritte umfasst:
    - Bilden (120) eines Punkts P^ = CRT(P (mod p), R (mod r 2)) in E^(Z/pr 2Z) in der Weise, dass sich P^ in E(Z/pZ) auf P reduziert und in
    Figure imgb0015
     auf R reduziert, wobei CRT das chinesische Restbildungsverfahren bezeichnet;
    - Berechnen (130) von Q^ = kP^ in E^(Z/pr 2Z) ;
    - Überprüfen (140), ob in
    Figure imgb0016
     ist und Ausgeben von Q = Q^ mod p nur dann, wenn dies der Fall ist.
  3. Verfahren nach Anspruch 1 oder 2, bei dem die ganze Zahl r zufällig gewählt wird.
  4. Verfahren nach Anspruch 1 oder 2, bei dem die ganze Zahl r einen vorgegebenden Wert besitzt.
  5. Verfahren nach einem der Ansprüche 3 und 4, bei dem die ganze Zahl r eine Primzahl ist.
  6. Verfahren nach Anspruch 2, bei dem der Punkt R in
    Figure imgb0017
     zufällig gewählt wird.
  7. Verfahren nach Anspruch 1 oder 2, bei dem die elliptische Kurve als eine Edwards-Kurve oder als eine Jacobi-Kurve dargestellt wird.
  8. Vorrichtung (200) zum Prüfen der Richtigkeit einer kryptographischen Operation auf einer ersten elliptischen Kurze E(Z/pZ), wobei die Vorrichtung (200) Mittel (220) umfasst zum:
    - Erhalten einer dritten elliptischen Kurze E^(Z/pr 2Z) ≡
    Figure imgb0018
     die durch eine chinesische Restbildung von der ersten elliptischen Kurve E(Z/pZ) und von einer zweiten elliptischen Kurze
    Figure imgb0019
     gegeben ist, wobei r eine ganze Zahl ist;
    - Ausführen der Operation an E^(Z/pr 2Z), um ein erstes Ergebnis zu erhalten;
    - Ausführen der Operation an
    Figure imgb0020
     wobei
    Figure imgb0021
     die Teilmenge der Punkte in
    Figure imgb0022
     bezeichnet, die Modulo r auf das Identitätselement auf
    Figure imgb0023
     reduzieren, um ein zweites Ergebnis zu erhalten;
    - Überprüfen, dass das erste Ergebnis und das zweite Ergebnis in
    Figure imgb0024
     gleich sind; und wenn das der Fall ist;
    - Ausgeben des ersten Ergebnisses der Operation in E^(Z/pr 2Z), Modulo p reduziert.
  9. Vorrichtung (200) nach Anspruch 8, bei der die kryptographische Operation eine Skalarmultiplikation in E(Z/pZ) für die fehlerresistente Berechnung von Q = kP auf einer elliptischen Kurve E(Z/pZ) ist, wobei die Vorrichtung (200) Mittel (220) umfasst zum:
    - Bilden eines Punkts P^ = CRT(P (mod p), R (mod r2)) in E^(Z/pr 2Z) in der Weise, dass sich P^ in E(Z/pZ) auf P reduziert und in
    Figure imgb0025
     auf R reduziert, wobei CRT das chinesische Restbildungsverfahren bezeichnet;
    - Berechnen von Q^ = kP^ in E^(Z/pr 2Z) ;
    - Überprüfen, ob in
    Figure imgb0026
     ist und Ausgeben von Q = Q ∧ mod p nur dann, wenn dies der Fall ist.
  10. Computerprogrammprodukt (240), in dem Anweisungen gespeichert sind, das, wenn es durch einen Prozessor ausgeführt wird, das Verfahren nach einem der Ansprüche 1 bis 7 ausführt.
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US8817974B2 (en) * 2011-05-11 2014-08-26 Nxp B.V. Finite field cryptographic arithmetic resistant to fault attacks
EP2535804A1 (de) * 2011-06-17 2012-12-19 Thomson Licensing Fehlerbeständiger Potenzierungsalgorithmus
CN102547694A (zh) * 2012-02-20 2012-07-04 上海电力学院 一种传感器网络中基于中国剩余定理的组密钥建立方法
FR2992084B1 (fr) * 2012-06-19 2015-07-24 Logiways France Procede de protection d'un circuit de cryptographie et systeme correspondant
JP6095584B2 (ja) * 2014-01-15 2017-03-15 日本電信電話株式会社 マルチパーティ計算システム、秘匿計算装置、マルチパーティ計算方法及びプログラム
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DE102016013692A1 (de) * 2016-11-16 2018-05-17 Giesecke+Devrient Mobile Security Gmbh Punktmultiplikation auf einer Erweiterung einer elliptischen Kurve
US10541866B2 (en) 2017-07-25 2020-01-21 Cisco Technology, Inc. Detecting and resolving multicast traffic performance issues
DE102017117899A1 (de) * 2017-08-07 2019-02-07 Infineon Technologies Ag Durchführen einer kryptografischen Operation
US10601578B2 (en) * 2017-10-26 2020-03-24 Nxp B.V. Protecting ECC against fault attacks
EP3503459B1 (de) * 2017-12-22 2021-04-21 Secure-IC SAS Vorrichtung und verfahren zum schutz der ausführung einer kryptographischen operation
CN110798305B (zh) * 2019-09-24 2023-05-30 瓦戈科技有限公司 一种故障分析防御方法、电子设备、可读存储介质

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CN100461668C (zh) * 2004-12-09 2009-02-11 中国电子科技集团公司第三十研究所 一种用于椭圆曲线密码算法芯片的倍点运算电路
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ATE536584T1 (de) 2011-12-15
US20100232599A1 (en) 2010-09-16
JP5528848B2 (ja) 2014-06-25
JP2010217880A (ja) 2010-09-30
EP2228716A1 (de) 2010-09-15
US8457303B2 (en) 2013-06-04
CN101840325B (zh) 2014-05-21
CN101840325A (zh) 2010-09-22

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